Hundreds of expatriates lined up at consulates in Germany, which has 1.4 million eligible Turkish voters, and in five other European countries ahead of the controversial April 16th plebiscite.
The vote sparked a bitter row when Germany and The Netherlands blocked campaign events by Turkish ministers earlier this month, leading a furious Erdogan to accuse both countries of using "Nazi" methods.
As in Turkey, the voters flocking to polling booths were divided on whether the proposal would help bring stable government or allow a strongman to impose one-man rule.
"I voted for democracy!" said one, Hussein Saregul, indicating that he had stamped the brown ballot paper for a "no" vote - not the white version for "yes".
Saregul, who has lived for eight years in the eastern city of Dresden with his family, said he was sorry relations between Germany and Turkey had been so tense in recent weeks.
"We hope that the 'no' vote will prevail," he said, adding that the referendum is only "in the interest of one man. It is a step towards dictatorship."
Germany, the most populous EU country, is home to the world's largest overseas Turkish community, about three million strong, a legacy of the "guest worker" programme of the 1960s and 1970s.
Turkish voters in Austria, Belgium, Denmark, France and Switzerland can also cast their ballots for the next two weeks, until April 9th.
Other countries will start later, with a total of around three million allowed to vote in 120 Turkish missions in 57 countries.
Turkey itself will vote on April 16th on the proposal to create an executive presidency and abolish the post of prime minister.
Germany and other Western nations have voiced concern about the plan, and about a crackdown in Turkey in the aftermath of a failed coup last July that has seen thousands of people arrested or fired from their posts.
Germany's top-circulation newspaper Bild recommended opposing the constitutional change, in a bilingual German-Turkish article.
It argued that modern Turkey's founder Mustafa Kemal Ataturk, "would have said no", describing him as "authoritarian but not dictatorial".
Turkish community leader Gokay Sofuoglu spoke of a "deep split" that ran through the diaspora and said the rift had intensified because Erdogan was labelling opponents of the referendum traitors and terrorists.
One voter in Berlin, Aslan Ismael, accused European countries of being biased against Erdogan.
"The German media and German politicians want a 'no' vote," he said, also complaining that Turkish ministers had been blocked from several rallies by local authorities citing logistical reasons.
"I don't understand this bashing, these anti-Erdogan measures," he said. "Those high-ranking officials wanted to come and explain what the reform is all about.
"In Germany, the principle of freedom of expression is very important and it has not been respected," he said, adding that he was not a member of Erdogan's Justice and Development Party (AKP).
An opponent of the proposed change, Berlin-born Sirin Manolya Sak, said that in the debate tensions "have been fuelled by both sides".
She said she regretted that Turks and Germans "who have lived together for 60 years in Germany - neighbours, friends, work colleagues - are arguing today".
"It's a shame ... that suddenly we wonder if some are integrated or not, just because they vote this way or that."